On the Mullineux involution for Ariki-Koike algebras

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On the Mullineux involution for Ariki-Koike algebras

Nicolas Jacon, Cédric Lecouvey

To cite this version:

Nicolas Jacon, Cédric Lecouvey. On the Mullineux involution for Ariki-Koike algebras. Journal of Algebra, Elsevier, 2009, 321 (No. 8), pp.2156-2170. �hal-00269840v3�

ALGEBRAS

NICOLAS JACON AND C´EDRIC LECOUVEY

Abstract. This note is concerned with a natural generalization of the Mullineux involution for Ariki-Koike algebras. By using a result of Fay- ers together with previous results by the authors, we give an efficient algorithm for computing this generalized Mullineux involution. Our al- gorithm notably does not involve the determination of paths in affine crystals.

1. Introduction

It is known that the set of simple modules of the symmetric group over a field of characteristic p can be labelled by the set Pp of p-regular parti- tions, that is, the partitions where no non zero part is repeated p or more times. The tensor product of one of these simple modules with the sign representation is again simple. This gives a natural involution I on Pp. In 1979, Mullineux [23] introduced a combinatorial algorithm yielding an in- volution m on P_p and conjectured the equality m= I. In 1994, Kleshchev [19] obtained the first combinatorial description of the involutionI by using sequences ofp-regular partitions. In particular, the corresponding algorithm kis different from the combinatorial proceduremof Mullineux. Finally, the conjecture m = I (and thus the equality m = k) was proved by Ford and Kleshchev in [14] (see also [6] and [8] for different proofs). For the sake of simplicity, we will refer to the involution I=m as the Mullineux involution in the sequel.

A natural quantum analogue of this problem has been considered by Richards [24] and by Brundan [7] in the context of the Hecke algebra of type A (see also [8] and [20]). The Mullineux involution can then be inter- preted as a skew-isomorphism ofA⁽¹⁾_e−1-crystals, that is as an isomorphism of oriented graphs switching the sign of each arrow. The resulting algorithm then coincides with that introduced by Kleshchev in [19].

In this note, we study an analogue of the Mullineux involution for the Ariki-Koike algebras (also called cyclotomic Hecke algebras of typeG(l,1, n) in the literature). Let l, n be positive integers, R be a field of arbitrary characteristic and consider (q, Q₀, Q₁, ..., Q_l−1) an l+ 1-tuple of invertible elements inRsuch thatq6= 1. The Ariki-Koike algebraHR,n :=HR,n(q;Q₀, Q₁, ..., Q_l−1) over R is the unital associativeR-algebra presented by:

Date: April, 2008.

1

• generators: T₀,T₁,...,T_n−1,

• relations:

T₀T₁T₀T₁ =T₁T₀T₁T₀,

TiTi+1Ti=Ti+1TiTi+1 (i= 1, ..., n−2), T_iT_j =T_jT_i (|j−i|>1),

(T₀−Q₀)(T₀−Q₁)...(T₀−Q_l−1) = 0, (T_i−q)(T_i+ 1) = 0 (i= 1, ..., n−1).

Ifl= 1 (resp. l= 2), we obtain a Hecke algebra of type A (resp. of type B). To study these algebras, by a theorem of Dipper and Mathas [9], it is sufficient to consider the case where Q₀ = q^s0, Q₁ = q^s1, ..., Q_l−1 = q^sl−1 for s = (s₀, s₁, ..., s_l−1) an l-tuple of integers. The algebra HR,n is then a cellular algebra in the sense of Graham and Lehrer [13]. As a consequence, we can define Specht modules which are indexed by thel-partitions of rank n. Recall that an l-partition λ of rank n is a sequence of l partitions λ= (λ⁽⁰⁾, ..., λ^(l−1)) such that

Xl−1

k=0

|λ^(k)| = n. We denote by Π_l,n the set of l- partitions of rankn. Hence, to each λ∈Π_l,n is associated a Specht module S^λ. Let ebe the multiplicative order ofq (we put e=∞ if q is not a root of unity). The Specht modules are in general not irreducible. Nevertheless, there exists a natural bilinear form,H_R,n-invariant, on each of these modules and an associated radical such that the quotients D^λ :=S^λ/rad(S^λ) are 0 or irreducible. Moreover, the non-zero D^λ provide a complete set of non isomorphic simple modules. Let Φ^s_e(n) be the set of l-partitions such that D^λ is non zero (where we sets= (s₀(mode), ..., s_l−1(mode))). It has been shown by Ariki [1] and Ariki-Mathas [4], that this set equals to the set of Kleshchevl-partitions. Thesel-partitions are generalizations of thee-regular partitions. They appear as the vertices of a distinguished realization of the abstractA⁽¹⁾_e−1-crystalB_e(Λs) associated to the irreducibleUv(slc_e)-module of highest weight Λs(see section 2.1).

Following [10], denote by HeR,n the algebra HR,n(q⁻¹;s_l−1, ..., s₀) and write Te₀,...,Te_l−1 for the standard generators of HeR,n. If λ∈Π_l,n, we write Se^λ for the corresponding Specht module on He_R,n and De^λ for the corre- sponding irreducible one. We have an isomorphismθ :HR,n →HeR,n given by

T₀7→ Te₀ T_i 7→ −qTe_i (i= 1, ..., n−1).

Put es = (−s_l−1(mod e), ...,−s0(mod e)). Then, θ induces a functor F_l from the category of HeR,n-modules to the category of HR,n modules. As a consequence, we obtain a bijective map

m_l: Φ^s_e→Φ^es_e,

satisfying

F_l(De^ml(λ))≃D^λ,

for allλ∈Φ^s_e. This map can be viewed as a generalization of the Mullineux involution. In [10], Fayers describes this generalized Mullineux involution by using skew-isomorphisms of A⁽¹⁾_e−1-crystals. This description is similar to a former one obtained in [21] for the analogue of the Mullineux involution, namely the Zelevinsky involution, in affine Hecke algebras of type A. Both rest on the determination of paths in crystal graphs. The aim of this note is to give an alternative efficient description of the map m_l in the spirit of the original procedure of Mullineux, that is without using paths in crystals.

We obtain an algorithm to compute the Mullineux involution for all positive integerl. This is achieved by combining the known description of the original Mullineux involution (i.e. for l= 1) with results of [17] on isomorphisms of A⁽¹⁾_e−1-crystals.

The following sections are structured as follows. Section 2 is devoted to a brief review onA⁽¹⁾_e−1-crystals and their labellings by Uglov and Kleshchev l-partitions. In Section 3, we mainly recall and reformulate in an appropriate language results of [16] and [17]. Section 4 contains the main result of this paper, namely the description of the mapm_l on Kleshchevl-partitions.

2. A⁽¹⁾_e−1-crystals and the Mullineux involution

Let Uv(csl_e) be the quantum group of affine typeA⁽¹⁾_e−1. This is an asso- ciative Q(v)-algebra with generators e_i, f_i, t_i, t⁻¹_i (for i = 0, ..., e−1) and

∂ and relations given in [25, §2.1]. We denote by U_v^′(slc_e) the subalgebra generated by e_i, f_i, t_i, t⁻¹_i (for i = 0, ..., e−1). We begin this section, by reviewing some background on the crystal graph theory of the irreducible highest weight U_v^′(slc_e)-modules. In particular, we recall the notion of Uglov and Kleshchevl-partitions and describe Fayers work on the Mullineux invo- lution. We refer to [18] and to [2] for the general theory of crystals. [12,§7]

gives a nice survey on some of the problems we will consider.

2.1. Realizations of abstract A⁽¹⁾_e−1-crystals. By slightly abuse of nota- tion, we identify the elements of Z/eZ with their corresponding labels in {0, ..., e−1} when there is no risk of confusion. Letv be an indeterminate and e > 1 be an integer. Write {Λ0, ...,Λ_e−1, δ} for the set of dominant weights of Uv(csl_e). Let l ≥ 1 ∈ N and consider s = (s₀, ..., s_l−1) ∈ Z^l. We set s = (s₀(mode), ..., s_l−1(mode)) ∈(Z/eZ)^l and Λs := Λ_s0(mode)+...+ Λ_sl−1(mode).

As a C(v)-vector space, the Fock space F_e of level l admit the set of all l-partitions as a natural basis. Namely the underlying vector space is

F_e=M

n≥0

M

λ∈Πl,n

C(v)λ,

where Π_l,n is the set of l-partitions with rank n. One can put different structures ofUv(slc_e)-modules on F_e. To describe them, we need some com- binatorics.

Let λ be an l-partition (identified with its Young diagram). Then, the nodes of λ are the triplets γ = (a, b, c) wherec ∈ {0, ..., l−1} and a, b are respectively the row and column indices of the node γ inλ^(c).The content ofγ is the integerc(γ) =b−a+s_c and the residue res(γ) ofγ is the element of Z/eZ such that

(1) res(γ)≡c(γ)(mod e).

We will say that γ is an i-node of λ when res(γ) ≡ i(mode). Finally, We say thatγ is removable whenγ = (a, b, c)∈λ and λ\{γ} is an l-partition.

Similarly γ is addable when γ= (a, b, c)∈/ λand λ∪ {γ}is anl-partition.

We now define distinct total orders on the set of addable and removable i-nodes of the multipartitions. Considerγ₁ = (a₁, b₁, c₁) andγ₂= (a₂, b₂, c₂) two i-nodes in λ. We define an order≺s by setting

γ₁≺sγ₂ ⇐⇒

c(γ₁)< c(γ₂) or

c(γ₁) =c(γ₂) andc₁ > c₂

Letλandµbe twoℓ-partitions of ranknandn+1 and assume there exists an i-nodeγ such that [µ] = [λ]∪ {γ}. We define the following numbers:

N_i^≻(λ,µ) =♯{addable i-nodesγ of λ such that γ^′ ≻sγ}

−♯{removable i-nodes γ^′ of µ such thatγ^′ ≻^sγ}, N_i^≺(λ,µ) =♯{addablei-nodes γ^′ ofλ such that γ^′ ≺sγ}

−♯{removablei-nodesγ^′ of µ such that γ^′ ≺sγ}, N_i(λ) =♯{addablei-nodes ofλ}

−♯{removablei-nodes ofλ}for 0≤i≤e−1 and M₀(λ) =♯{0-nodes ofλ}.

The following theorem is [25, Thm. 2.1].

Theorem 2.1 (Jimbo et al. [15], Foda et al. [11], Uglov [25]). Let s ∈Z^l. Then we can put an action of Uv(csl_e) on Fe as follows.

eiλ= X

res([λ]/[µ])=i

q^{−Ni≻(µ,λ)}µ, fiλ= X

res([µ]/[λ])=i

q^Ni≺(λ,µ)µ,

t_iλ=q^Ni(λ)λ, ∂λ=−(∆ +M₀(λ))λ, (0≤i≤e−1) where∆is a rational number defined in[25, Thm 2.1]which does not depend on λ(but depends ons). The associatedUv(csl_e)-module which is denoted by F_e^s is an integrable module.

Now, one can define another structure ofUv(slc_e)-module on the Fock space Fe. This is achieved by considering the following order on the set ofi-nodes of anl-partition :

γ1 ≺sγ2⇐⇒

c1> c2 or

c₁=c₂ and c(γ₁)< c(γ₂) .

Using the same formulas as in theorem 2.1 where ≺^s is replaced by ≺^s and

∆ = 0, the space Fe is endowed with a different structure of an integrable Uv(slc_e)-module that we will denote by F_e^s. We refer to [2, §10.2] for the proof of this assertion.

Observe that the order ≺s and thus F_e^s really depends on s whereas ≺s

and thus F_e^s only depends on the classs∈(Z/eZ)^l. Remark 2.2.

(1) For l= 1, the Fock spacesF^s_e and F^s_e coincide.

(2) With s= (s₀(mode), ..., s_l−1(mode))), we have (2) F^s_e=F^s_e⁰ ⊗ · · · ⊗F^s_e^l−1.

The Fock space F^s_e is a tensor product of Fock spaces of level 1.

The Uv(csl_e)-submodule of F^s_e generated by the empty l-partition is de- noted by V_e^s(Λs). By [25, Rem. 2.2], this is an irreducible highest weight U_v(csl_e)-module with weight Λs−∆δ. Similarly, theU_v(csl_e)-submodule ofF^s_e generated by the emptyl-partition is denoted byV_e^s(Λs). By [2, thm 10.10], this is an irreducible highest weightUv(csl_e)-module with weight Λs.

We rather need in the sequel the restrictionsV_e^s(Λs)^′ andV_e^s(Λs)^′ ofV_e^s(Λs) and V_e^s(Λs) to the subalgebraU_v^′(slc_e). They are both irreducible asU_v^′(slc_e)- modules of highest weight Λs. In particular, all the modules V_e^s(Λs)^′,s∈s are isomorphic to V_e^s(Λs)^′. However, as explained above, the correspond- ing actions of the Chevalley operators on these modules do not coincide in general.

The modules F^s_e and F^s_e are integrable. Hence, by the general theory of Kashiwara crystal bases, F^s_e and F^s_e have crystal graphs that we denoteB_e^s and B_e^s,respectively. Since the module structuresF^s_e and F^s_e depend on the total orders≺s and ≺^s, the graph structures onB_e^sand B_e^sdo not coincide in general although they both admit the set ofl-partitions as set of vertices.

To describe the graph structure of B_e^s (for a fixed s ∈ s) we proceed as follows. Starting from any l-partition λ, we can consider its set of addable and removable i-nodes. Let w_i be the word obtained first by writing the addable and removable i-nodes of λ in increasing order with respect to ≺s

next by encoding each addable i-node by the letter A and each removable i-node by the letter R. Write we_i = A^pR^q for the word derived from w_i by deleting as many of the factors RA as possible. If p > 0, let γ be the rightmost addablei-node inwe_i. Whenwe_i 6=∅, the nodeγ is called the good i-node. Now, the crystalB_e^sis the graph with

• vertices: the l-partitions,

• edges: λ →ⁱ µ if and only if µ is obtained by adding to λ a good i-node.

The graph structure on and B_e^s is obtained similarly by using the order≺s

instead of ≺^s.

2.2. Uglov and Kleshchev multipartitions. By definition V_e^s(Λs) and V_e^s(Λs) are the irreducible components with highest weight vector ∅ in F^s_e and F^s_e, their crystal graphs B_e^s(Λs) and B_e^s(Λs) (which are also the crystal graphs ofV_e^s(Λs)^′andV_e^s(Λs)^′) can be realized respectively as the connected components of highest weight vertex ∅inB_e^s andB_e^s.

Example 2.3. The graph below is the subgraph of B₄^(0,0,1)(2Λ₀+ Λ₁) con- taining the 3-partitions of rank less or equal to 4.

(∅,∅,∅)

(1,∅,∅) (∅,∅,1)

(2,∅,∅) (1,1,∅) (1,1,∅)

(3,∅,∅) (2,∅,1) (2,1,∅) (1,∅,2) (∅,∅,3)

(4,∅,∅) (3,1,∅) (2,∅,2)(2.1,∅,1) (2,2,∅) (1,1,2) (1,∅,3) (∅,∅,4)

HH HHj

@@R @

@ R

@@R @

@

R @

@ R

A

@ AU

@

R ?

QQ Q

s @

@ R

0 1

0 0

1

1 1

1

0 1

0 0

1 1

1 0 0 1

We denote by Φ^s_e the set of vertices in B_e^s(Λs). The elements of Φ^s_e are called the Uglov l-partitions. We also write Φ^s_e(n) for the subset of Φ^s_e containing the Uglov l-partitions with rankn. In general the set Φ^s_e is very difficult to describe without computing the underlying crystal. Nevertheless, in the case where 0≤s₀ ≤ ··· ≤s_l−1 ≤e−1,Foda et al. have given a simple non recursive description of the Uglov l-partitions (see [11, Thm 2.10]).

We denote by Φ^s_e the set of vertices in B^s_e(Λs). The elements of Φ^s_e are called the Kleshchev l-partitions. We define Φ^s_e(n) as we have done in the Uglov case. Note that there exists only a recursive description of the Kleshchevl-partitions Φ^s_e,except when the number of fundamental weights appearing in the decomposition of Λs is less or equal to 2 , see [3] and [5].

In the other cases, they can only be obtained by computing the underlying crystal or using the procedure described in [17, §5.4].

Example 2.4. The graph below is the subgraph ofB₄^s(2Λ₀+Λ₁) withs= (0 (mod 4),0 (mod 4),1 (mod 4)) containing the 3-partitions of rank less than 4.

(∅,∅,∅)

(∅,1,∅) (∅,∅,1)

(∅,1,1) (1,1,∅) (∅,∅,2)

(∅,1,2) (∅,2,1) (1,1,1) (∅,∅,2.1) (∅,∅,3)

(∅,1,3) (1,1,2) (∅,2,2)(∅,2.1,1) (1,2,1) (∅,1,2.1)(∅,∅,3.1)(∅,∅,4)

HH HHj

@@R @

@ R

@@R @

@

R @

@ R AAU

@@R ? QQ

Q

s @

@ R

0 1

0 0

1

1 1

1

0 1

0 0

1 1

1 0 0 1

Let s^′ := (s^′₀, ..., s^′_l−1) ∈ Z^l be such that s^′ ∈ s. Then the two modules V_e^s(Λs)^′ andV_e^s′(Λs) are both isomorphic toV_e^s(Λs)^′. Hence the correspond- ing three crystal graphs are isomorphic. These crystal isomorphisms define the following bijections

Ψ^s_e,n^→s′ : Φ^s_e(n)→Φ^s_e^′(n) Ψ^s_e,n^→s: Φ^s_e(n)→Φ^s_e(n).

As it can be easily checked on the previous examples, the sets Φ^s_e(n) and Φ^s_e(n) do not coincide in general. However, if we fixn, it is possible to realize the set Φ^s_e(n) of Kleshchevl-partitions with ranknas a special set Φ^s_e^∞(n), s∞∈sof Uglov l-partitions. We refer the reader to Lemma 5.4.1 in [17] for the proof of the following proposition.

Proposition 2.5. Let n ∈ N and consider s∞ := (s₀, ..., s_l−1) ∈ Z^l such that

(3) s_i+1−s_i > n−1

for all i= 0, ..., l−2. ThenΨ^s_e,n^∞→s is the identity. Hence, we have : Φ^s_e(n) = Φ^s_e^∞(n).

In particular, Φ^s_e^∞(n) does not depend on the multicharge s_∞ provided (3) is satisfied.

This shows that the description of the bijections Ψ^s_e,n^→s′ for all s,s^′ ∈ s will lead that of Ψ^s_e,n^→sas a special case. A multicharge verifying (3) is called asymptotic (this is called n-dominant in [26]). This justifies the notation Φ^s_e^∞(n) we have adopted.

3. Isomorphisms ofA⁽¹⁾_e−1-crystals

We now review some results obtained in [16] and [17] making explicit the bijections Ψ^s_e,n^→s′. In [17], we have chosen to explain our results in the

language of column shaped tableaux to make apparent the link with the combinatorics of sl_n. Here, we rather adopt the equivalent language of symbols also used in [16] which permits to give a very simple description of the elementary steps involved in our algorithms.

3.1. Elementary transformations in Sb_l. We write as usual Sb_l for the extended affine symmetric group in type A_l−1.This group is generated by the elements σ₁, ..., σ_l−1 and y₀, ...., y_l−1 together with the relations

σ_cσ_c+1σ_c =σ_c+1σ_cσ_c+1, σ_cσ_d=σ_dσ_c for |c−d|>1, σ_c² = 1, with all indices in {1, ..., l−1}and

y_cy_d=y_dy_c, σ_cy_d=y_dσ_c ford6=c, c+ 1, σ_cy_cσ_c =y_c+1.

For any c ∈ {0, ..., l−1}, we set z_c = y₁· · ·y_c. Write also ξ =σ_l−1· · ·σ₁ andτ =y_lξ.Sincey_c =z_c−1⁻¹ z_c,Sb_l is generated by the transpositionsσ_c with c∈ {1, ..., l−1}and the elementsz_c withc∈ {1, ..., l}.Observe that for any c∈ {1, ..., l−1},we have

(4) z_c =ξ^l−cτ^c.

This implies thatSb_l is generated by the transpositionsσ_c withc∈ {1, ..., l− 1} andτ.Consider ea fixed positive integer. We obtain a faithful action of Sb_l on Z^l by setting for anys= (s₀, ..., s_l−1)∈Z^l

σ_c(s) = (s₀, ..., s_c, s_c−1, ..., s_l−1) andy_c(s) = (s₀, ..., s_c−1, s_c+e, ..., s_l−1).

Then τ(s) = (s1, s2, ..., s_l−1, s0+e).

Remark 3.1. ¿From the preceding considerations, {s1, . . . , s_l−1, τ} is a min- imal set of generators for Sb_l. Hence, the bijections Ψ^s_e,n^→s′ : Φ^s_e(n) →Φ^s_e^′(n) with s∈s and Ψ^s→_e,n^s: Φ^s_e(n)→Φ^s_e(n) can be obtained by composing bijec- tions corresponding to the casess^′=τ(s) ands^′=σ_c(s) withc= 1, ..., l−1.

3.2. Elementary crystals isomorphisms. The following proposition is stated in [17, Prop. 5.2.1]. We give the proof for the convenience of the reader.

Proposition 3.2. Let s ∈Z^l and e∈N. Let λ= (λ⁽⁰⁾, ..., λ^(l−1))∈Φ^s_e(n).

Then

Ψ^s_e,n^→τ(s)= (λ⁽¹⁾, ..., λ^(l−1), λ⁽⁰⁾).

Proof. Set s = (s₀, ..., s_ℓ−1). Then τ(s) = (s₁, ...., s_ℓ−1, s₀ +e). Consider λ = (λ⁽⁰⁾, ..., λ^(ℓ−1)) a multipartition and set λ^♦ = (λ⁽¹⁾, ..., λ^(ℓ−1), λ⁽⁰⁾).

Consideri∈ {0,1, ..., e−1}andγ₁ = (a₁, b₁, c₁), γ₂ = (a₂, b₂, c₂) twoi-nodes of λ. Then γ^♦₁ = (a₁, b₁, c₁−1(mode)) and γ₂^♦ = (a₂, b₂, c₂ −1(mod e)) are two i-nodes of λ^♦. We then easily check that γ₂ ≺^s γ₁ if and only if γ^♦₂ ≺_τ(s)γ₁^′.This implies that Ψ^s→τ(_e,n ^s)(λ) =λ^♦.

The description of the bijections Ψ^se,n^→σc(s), c= 1, ..., l−1 are more com- plicated. It essentially rests on the following basic procedure on pairs (U, D) of strictly increasing sequences of integers. let U = [x₁, . . . , x_r] and D = [y₁, . . . , y_s] two strictly increasing sequences of integers. We compute from (U, D) a new pair (U^′, D^′) with U^′ = [x^′₁, . . . , x^′_r] and D^′= [y₁^′, . . . , y^′_s] of such sequences by applying the following algorithm :

Algorithm.

• Assume r ≥s. We associate toy1 the integer xi1 ∈U such that

(5) x_i1 =

max{x∈U |y₁ ≥x} ify₁ ≥x₁

x_r otherwise .

We repeat the same procedure to the ordered pair (U \ {x_i1}, D\ {y1}).By induction this yields a subset{xi1, . . . , x_is} ⊂U.Then we define D^′ as the increasing reordering {xi1, . . . , xis} and U^′ as the increasing reordering of U \ {xi1, . . . , x_is} ⊔D.

• Assume r < s.We associate to x1 the integer yi1 ∈D such that

(6) y_i1 =

min{y ∈D|x₁ ≤y}ifx₁≤y_s

y₁ otherwise .

We repeat the same procedure to the ordered sequencesU\{x1}and D\ {yi1} and obtain a subset{yi1, . . . , y_ir} ⊂D.Then we defineU^′ as the increasing reordering {y_i1, . . . , y_ir} and D^′ as the increasing reordering of D\ {yi1, . . . , y_ir} ⊔U.

Example 3.3. Take U = [0,1,3] and D = [0,1,2,4,7]. The subset of U determined by the previous algorithm is {0,1,4}. This gives U^′ = [0,1,4]

and D^′ = [0,1,2,3,7].

Now consider c ∈ {1, . . . , e−1} and λ = (λ⁽⁰⁾, ..., λ^(l−1)) ∈ Φ^s_e(n). In order to compute Ψ^s→σe,n ^c(s)(λ), we need the symbol Sc of the bipartition (λ^(c−1), λ^(c)). Letd_c−1 and d_c be the number of nonzero parts inλ^(c−1) and λ^(c) and put m= max(d_c−1−s_c−1, d_c−s_c) + 1.Set

( β_i^(c−1) =λ^(c−1)_i −i+s_c−1+mfor any i= 1, . . . , m+s_c−1 β_i^(c)=λ^(c)_i −i+s_c+m for any i= 1, . . . , m+s_c

where, by convention the partitionsλ^(c−1)andλ^(c)are taken with an infinite number of zero parts. The symbol S_c is the two-row tableau

S_c = β^(c)_m+sc β_m+s^(c) _c−1 ... ... ... β₁^(c) β^(c−1)_m+sc−1 β_m+s^(c−1)_c−1−1 ... ... β₁^(c−1)

!

= U

D

Observe that β_i^(k), k ∈ {c−1, c} is nothing but the content of the node γ = (i, λ^(k)_i , k)∈λtranslated by m. This translation normalizes the symbol S_c so that β_m+s^(c) _c = β_m+s^(c−1)_c−1 = 0. In fact, for a fixed pair (s_c−1, s_c) and m as above, the map Σ_(sc−1,sc) which associates to each bipartition its symbol

is bijective. It is easy, from a symbol S_c, to recover the unique bipartition (λ^(c−1), λ^(c)) such that S_c is the symbol of (λ^(c−1), λ^(c)).We are now ready to describe the bijection Ψ^se,n^→σc(s): Φ^s_e(n)→Φ^σe^c(s)(n).

Let s ∈ Z^l and e ∈ N. Consider λ = (λ⁽⁰⁾, ..., λ^(l−1)) ∈ Φ^s_e(n) and c ∈ {1, ..., l−1}. WriteS_c= _D^U

for the symbol corresponding to the bipartition (λ^(c−1), λ^(c)). Denote by Se_c = ^U_D^e_e

the symbol obtained by applying to S_c the previous algorithm. Then we compute the bipartition (eλ^(c−1),λe^(c)) = Σ⁻¹_(s

c−1,sc)(Se_c). Finally we consider thel-partition λe = (λ⁽⁰⁾, . . . ,λe^(c),λe^(c−1). . . , λ^(l−1)) obtained by replacing inλ, λ^(c−1) by eλ^(c) and λ^(c) by eλ^(c−1). Proposition 3.4. With the above notation, we have

Ψ^s→σ_e,n ^c−1(s)(λ) =λ.e

Proof. The above algorithm is the translation in terms of symbols of Propo-

sition 5.2.2 in [17].

Example 3.5. Let e = 3 and let s = (0,2,0). Then one can check that the 3-partition (4.1,3.1,1) belongs to Φ^s_e(10). We want to determine the 3- partition Ψ^s_e,n^→σ2(s)(4.1,3.1,1). We first write the symbolS₂ of (3.1,1) with respect to s and m= 3 :

S2 =

0 1 3 0 1 2 4 7

.

By using the previous example, we obtain Se₂ =

0 1 4 0 1 2 3 7

.

This thus gives eλ^(c) = (2) and eλ^(c−1) = (3). Hence Ψ^s→σe,n ^2(s)(4.1,3.1.1,∅) = (4.1,2,3).

4. Mullineux involution for Kleshchev l-partitions

4.1. Mullineux involution as a skew crystal isomorphism. We here keep the setting of the introduction and review a result of Fayers [10] which shows how the Mullineux involution on Kleshchev l-partitions can be com- puted by using paths in crystal graphs. The generalized Mullineux involu- tion can be regarded as a bijection

m_l: Φ^s_e→Φ^es_e

wheres= (s₀(mode), ..., s_l−1(mode)) andes= (−sl−1(mode), ...,−s0(mod e)).

Theorem 4.1 (Fayers [10]). Let λ∈Φ^s_e and consider in B_e^s(Λs) a path

∅→ ·ⁱ¹ → ·ⁱ² → · · ·ⁱ³ →ⁱⁿ λ

from the empty l-partition toλ.There exists a corresponding path

∅⁻ⁱ→ ·^{1 −i}→ ·^{2 −i}→ · · ·^{3 −i}→ⁿµ

inB^es_e(Λs)from the emptyl-partition to anl-partitionµ∈Φ^es_e. We have then m_l(λ) =µ.

In the level 1 case (i.e. forl= 1), the mapm₁ can be described indepen- dently of the notion of crystal graph by using an algorithm originally due to Mullineux. We refer for example to [14] for a complete exposition of this procedure. For all s∈Z, it gives the map

m₁: Φ^s_e→Φ⁻_e^s.

wheres=s(mode) Our aim is now to show how the generalized Mullineux involution (i.e. in level l >1) can be computed from the map m1 and the results of Section 3. For anys= (s₀(mode), ..., s_l−1(mode))∈(Z/eZ)^l, set

−s= (−s0(mode), ...,−s_l−1(mod e)), es= (−sl−1(mode), ...,−s0(mode)).

Our strategy is as follows. Letλbe in Φ^s_eand consider a path from the empty l-partition to λinB_e^s(Λs) with arrows successively labelled by i₁, ..., i_n.

(1) We first describe thel-partition ν in Φ⁻_e^sdefined by considering the path

∅⁻ⁱ→ ·^{1 −i}→ ·^{2 −i}→ · · ·^{3 −i}→ⁿν in B_e^−s(Λ−s).

(2) Next, we describe the bijection Φ⁻_e^s→Φ^es_edefined by the crystal iso- morphism betweenB_e^−s(Λ−s) andB^e_e^s(Λ−s).This permits to compute µ fromν.

4.2. Computing ν from λ.

Proposition 4.2. Let λ∈Φ^s_e and consider inB_e^s(Λs) a path

∅→ ·ⁱ¹ → ·ⁱ² → · · ·ⁱ³ →ⁱⁿ λ

from the empty l-partition toλ.Then there exists a corresponding path

∅⁻ⁱ→ ·^{1 −i}→ ·^{2 −i}→ · · ·^{3 −i}→ⁿν

in B_e^−s(Λ−s) from the empty l-partition to anl-partitionν ∈Φ⁻_e^s. We have then

ν = (m1(λ⁰), . . . , m1(λ^l−1)).

Proof. We know by (2) that the Fock spaces F^s_e and F⁻_e^s can be regarded as tensor products of level 1 Fock spaces. Therefore, each l-partition b = (b⁽⁰⁾, . . . , b^(l−1)) in the crystals B_e^s and B_e^−s of the Fock spaces F^s_e and F⁻_e^s can be written b=b⁽⁰⁾⊗ · · · ⊗b^(l−1).Recall that in Kashiwara crystal basis theory, the action of each crystal operator ee_i,fe_i, i ∈ {0, . . . , l−1} on b = b₀⊗· · ·⊗bl−1is determined by the intergersε_i(b_k) = max{r∈N|ee^r_i(b_k)6= 0}

and ϕ_i(b_k) = max{r |fe_i^r(b_k)6= 0} whenk runs over{0, . . . , l−1}.

We now proceed by induction on n. For n = 0, the proposition is clear.

Assume that the result is proved for n −1 and set λ=fe_in· · ·fe_i1(∅) in B_e^s(Λs). Consider λ♭=fein−1· · ·fei1(∅) ∈B_e^s(Λs) and ν♭=fe−in−1· · ·fe−i1(∅) ∈ B⁻_e^s(Λ−s). By induction, we have thenν♭ = (m₁(λ⁽⁰⁾♭ ), . . . , m₁(λ^(l−1)♭ )).The Mullineux involutionm₁switches the signs of the arrows. Thusε_i(m₁(λ^(k)♭ )) = ε_−i(λ^(k)_♭ ) and ϕ_i(m₁(λ^(k)_♭ )) = ϕ_−i(λ^(k)_♭ ) for any k = 0, . . . , l −1. Since fe_in(λ♭) = λ6= 0, we have fe_in(ν♭) =ν 6= 0 by the above arguments. More- over fe_in and fe_−in act on the same component of λ♭ and ν♭ considered as tensor products. Namely, there exists an integer a∈ {0, . . . , l−1}such that

( λ= (λ⁽⁰⁾♭ , . . . ,fein(λ^(a)♭ ), . . . , λ^(l−1)♭ )

ν = (m₁(λ⁽⁰⁾_♭ ), . . . ,fe_−in(m₁(λ^(a)_♭ )), . . . , m₁(λ^(l−1)_♭ )) .

We have m₁(fe_in(λ^(a)_♭ )) =fe_−in(m₁(λ^(a)_♭ )) since m₁ switches the sign of each arrow. This gives ν = (m₁(λ⁰), . . . , m₁(λ^l−1)) which establishes the propo-

sition by induction.

4.3. Computingµfromν. Now, assume we have computed the Kleshchev l-partitionν ∈Φ⁻_e^sfrom the Kleshchevl-partitionλ∈Φ^s_eas in the previous section. To computeµ∈Φ^es_e from the required crystal isomorphism, we need to realize ν as an Uglov l-partition. To do this, we consider an asymptotic multicharge −s ∈ −s. The multicharge s can be written s = (s₀ (mod e), . . . , s_l−1 (mod e)) with s_c in{0, . . . , e−1}for any c= 0, . . . , l−1. Now for any c = 1, . . . , l−1, write pc for the minimal nonnegative integer such that

sc−1−sc+pce > n−1.

Put then :

(7) −s:= (−s0,−s1+p₁e,−s2+p₁e+p₂e, . . . ,−sl−1+ Xl−1

i=1

p_ie).

Since −sis asymptotic, we have ν ∈Φ⁻_e^s by Prop 2.5 and Ψ⁻_e,n^s→−s(ν) =ν.

Now, with the notation of Section 2.1 for the generators of the affine Weyl group of type A, we set

w_(c,d)=σ_cσ_c−1...σ_d for anyc≥d >0.

Givenp∈ {1, ..., l−1}, we also introduce

γ_p =w_(l−p,1)...w_{(l−2,p−1)}w_(l−1,p).

Note that for v:= (v₀, ..., v_l−1)∈Z^l andp= 1, ..., l−1, we have:

γ_p.v= (v_p, ..., v_l−1, v₀, ..., v_p−1).

For each p= 1, ..., l−1, set

α_p=τ^l−pγ_p. We have thus, forv= (v₀, ..., v_l−1)∈Z^l

α_p.v= (v₀, ..., v_p−1, v_p+e, ..., v_l−1+e).

Finally, letw0 =w_(1,1)w_(2,1)...w_(l−1,1) be the longest element of the symmet- ric groupS_l and put

(8) η:=α^2(p₁ ^l−1+1)...α^2(p_l−1¹⁺¹⁾w₀.

Proposition 4.3. Consider the multicharge −s of (7) and set (9) es= (se₀, . . . ,sg_l−1) :=η.(−s)

Then for any i= 0, . . . , l−1, we have e

s_i≡ −sl−i−1(mod e).

Thereforees belongs toes. Moreover, the multichargees is asymptotic.

Proof. By definition ofes= (se₀, ...,sg_l−1), we have for all d= 0, . . . , l−2 : (10) sg_d+1−se_d=−sl−d−2+s_l−d−1+p_l−d−1e+ 2e.

Now, since the integers s_c, c= 0, . . . , ℓ−1 belong to{0, . . . , e−1}, we can write

−sl−d−2+s_l−d−1+ 2e≥s_l−d−2−s_l−d−1.

Hence, we derive from (10) and the definition of the integers p_d that g

s_d+1−se_d≥s_l−d−2−s_l−d−1+p_l−d−1e > n−1.

Thus, the multichargeη.(s) is asymptotic which proves our proposition.

Recall that we have a Kleshchevl-partitionν ∈Φ⁻_e^s. Using Section 3, we can compute

b

ν := Ψ⁻_e,n^s→es(ν)

wherees is defined by (9). Sincees is asymptotic and belongs toes,we derive from Proposition 2.5

Ψ^es_e,n^→es(νb) =bν.

Let us summarize the transformations we have computed fromλ∈Φ^s_e. (11) λ∈Φ^s_e^Ψ

s→−s

e,n→ ν ∈Φ⁻_e^sΨ

−s→−s

e,n→ ν ∈Φ⁻_e^sΨ

−s→es

e,n→ νb∈Φ^es_e^Ψ

−es→es e,n→ bν∈Φ^es_e

Recall now that we have a path

∅ →ⁱ¹ .→ⁱ² .→ⁱ³ ...→ⁱⁿ λ inB^s_e. By (11), we have the path

∅⁻ⁱ→¹ .⁻ⁱ→² .⁻ⁱ→³ ...⁻ⁱ→ⁿ bν inB^es_e. This gives

m_l(λ) =µ=νb

Hence, the procedure (17) gives a purely combinatorial algorithm for com- puting the Mullineux involution which does not use the crystal (and thus the notion of good nodes) of irreducible highest weight affine modules.

4.4. The case e = ∞. We now briefly focus on the case e = ∞. Our algorithm considerably simplifies because in this case, the Kleshchev and FLOTW l-partitions coincide. Moreover, the Mullineux involution m₁ re- duces to the ordinary conjugation of partitions. Consider λ ∈ Φ^s_∞. The computation ofm_l(λ) can be described as follows :

• First, by Proposition 4.2, the skew-isomorphism of crystals from B_e^s(Λs) to B_e^−s(Λ−s) which switches the sign of the arrows sends λ to

µ= ((λ⁰)^′, ...,(λ^l−1)^′)

where m₁ just corresponds to the usual conjugation of partitions.

• Next, w₀ being the longest element of the symmetric group, we have w0.(−s0, ...,−s_l−1) = (−s_l−1, ...,−s0).

Thus we derive

m_l(λ) =ν := Ψ^s_n^→w0.s(µ).

4.5. Concluding remarks. The algorithm of Section 3.2 also permits to compute the bijections Ψ^se,n^→σ(s) between Φ^s_e and Φ^σ(e ^s) for any σ ∈ Sn. By decomposing σ in terms of the generators s_c, c = 1, . . . l−1, it suffices to compute the isomorphisms Ψ^s_e,n^→sc(s). Since B_e^s can be regarded as a tensor product of crystals, we can assumel= 2 and describe the bijection

Ψ⁽_e,n^{s0,s1)→(s1,s0)}: Φ⁽_e^s0,s1)→Φ⁽_e^s1,s0).

Considerλ∈Φ⁽e^s0,s1) and choose (a0, a1) an asymptotic charge witha0 ∈s₀ and a₁ ∈s₁. We haveλ∈Φ^(ae ^0,a1). Then one computes Ψ^(ae,n^{0,a1)→(a1,a0)}(λ) = ν as in Proposition 3.4. Now the problem is that (a₁, a₀) is not asymptotic in general. To make it asymptotic, we need to compose translations byein Z² acting on the right coordinate. Observe that for any (v₀, v₁) ∈ Z², we haveτ s1(v0, v1) = (v0, v1+e).So if we setκ=τ s1, we have to compute the bipartitions

(12) µ^(k)=Ψ^(a_e,n^{1,a0)→τk(a1,a0)}(ν)

with k ∈ {0, . . . , m} where m ∈ N is minimal such that a₁ −a₀ +me >

n−1. Nevertheless, in practice one can restrict our computations and take m minimal such µ^(m) =µ^(m+1). This is because the Kleshchev bipartition is fixed by the bijection Ψ⁽_e,n^{s1,s0)→τ(s1,s0)}. Finally, one has

Ψ⁽_e,n^{s0,s1)→(s1,s0)}(λ) =µ^(m).

Such a stabilization phenomenon also happens during the computation of µfrom ν (see Section 4.3). This means that, in practice, we have

Ψ⁻_e,n^s→η(−s)(ν) = Ψ⁻_e,n^{s→η♭(−s)}(ν)

where η := α^2(p₁ ^l−1+1)· · ·α^2(p_l−1¹⁺¹⁾w₀ is defined as in (8) and where η^♭ :=

α^2(p

♭ l−1+1)

1 · · ·α^2(p_l−1^♭1+1)w₀ withp^♭_c ≤p_c forc∈ {0, . . . , l−1}.

The previous description of the bijections Ψ^s_e,n^→σ(s), σ ∈ S_l yields an al- ternative for computing Ψ⁻_e,n^s→es(ν) =µin Section 4.3.Nevertheless, one can verify that the number of elementary transformations it requires (i.e. the number of bijections Ψ^se,n^→sc(s), c= 1, . . . , l−1 or Ψ^se,n^→τ(s) to compute) is in general greater than the number of transformations used in Ψ^{−s→η(−s}_e,n ⁾. Example 4.4. We conclude this paper with a computation of the Mullineux involution on a 3-partition. Takee= 4 ands= (0(mod 4),1(mod 4),3(mod 4)).

Then, one can check that the 3-partition λ := (4,3,1.1.1)) is a Kleshchev 3-partition of rankn= 10 associated to s. Now we have

m1(4) = (2.1.1), m1(3) = (1.1.1), m1(1.1.1) = (3).

Hence, to λ, one can attach the Kleshchev 3-partition ν = (2.1.1,1.1.1,3) which is Kleshchev for −s = (0(mod 4),−1(mod 4),−3(mod 4)). Since p1 = p2 = 3, This 3-partition is an Uglov 3-partition associated with the multicharge

−s= (0,−1 + 3×4,−3 + 3×4 + 3×4) = (0,11,21).

Following§4.3, we put :

η =α⁸₁α⁸₂w₀. We compute

Ψ⁻_e,n^{s→η.(−s)}(ν) = (∅,1.1.1,4.1.1.1).

Hence, we obtain

m₃(λ) = (∅,1.1.1,4.1.1.1).

Acknowledgments. The authors are grateful to A. Kleshchev for useful comments and for having pointed out several references. Part of this work has been written while the authors were visiting the Mathematical Research Institute of Mathematics in Berkeley for the programs “Combinatorial Rep- resentation Theory” and “Representation Theory of Finite Groups and Re- lated Topics”. The authors want to thank the organizers of these programs and the MSRI for financial supports.